Linear difference equations form a very special but important class of dynamical systems. The theory of linear difference equations is well developed and provides a complete characterization of the solutions of these equations. In the present chapter, we discuss some elements of this theory that are of high relevance for economists. The importance of linear difference equations stems, on the one hand, from numerous applications in many areas and, on the other hand, from the fact that under certain conditions the solutions to non-linear difference equations can be approximated by those of suitably linearized equations. Before we turn to non-linear equations in chapter 3, we must therefore deal with the linear case.

We start in section 2.1 by discussing the difference between homogeneous and non-homogeneous linear difference equations and by showing that the set of trajectories of a homogeneous linear difference equation forms a finite-dimensional vector space. This allows us to represent every trajectory of such an equation as a linear combination of finitely many basis solutions. In section 2.2, we derive explicit formulas for the basis solutions of homogeneous linear difference equations with constant coefficients, and in section 2.3 we deal with the important special case of a two-dimensional system domain. Finally, in section 2.4 we present two different approaches for solving non-homogeneous linear difference equations.

Terminology and general results

Throughout this chapter, we assume that the system domain X is the entire n-dimensional Euclidean space ℝn. A difference equation with system domain X is said to be linear if it is of the form

xt+1 = A(t)xt + b(t) (2.1)

where A : ℕ0 → ℝn × n and b : ℕ0 → ℝn are a matrix-valued function and a vector-valued function, respectively. In words, the difference equation xt+1 = f(xt, t) is linear if the law of motion f is an (affine) linear function of the system variables.